Orthogonal

Perpendicular.

• $\vec{a}$ and $\vec{b}$ are orthogonal if $\vec{a}\cdot \vec{b}=0$

• $||\vec{u}+\vec{w}|{|}^2=||\vec{u}|{|}^2+||\vec{w}|{|}^2\vee \vec{u}\cdot \vec{w}=0⟹\text{Orthogonal}$
• If $W$ is a subspace of ${\mathbb{R}}^n$ and $\vec{z}\in {\mathbb{R}}^n$, $\vec{z}$ is orthogonal to $W$ if it is orthogonal to every vector in $W$
• The set of all vectors orthogonal to a subspace is a itself a subspace, called the Orthogonal Complement of $W$, $W^{\perp }$, W perp
  • $W^{\perp }=\{\vec{z}\in {\mathbb{R}}^n\mid \forall (\vec{w}\in W)\vec{z}\cdot \vec{w}=0\}$
• $dim(RowA)=dim(ColA)$
• $(RowA{)}^{\perp }=NulA$
• $(ColA{)}^{\perp }=Nul(A^T)$

$\vec{x}\in Nul(A)⟺$

• $A\vec{x}=\vec{0}$
• $\vec{x}$ is orthogonal to each row of $A$
• $RowA$ is orthogonal complement to $NulA$
• $dim(RowA)+dim(NulA)=$ number of columns

$\vec{a}\cdot \vec{b}=0\wedge \vec{a}\ne \vec{0},\vec{b}\ne 0⟺\vec{a}\perp \vec{b}⟺a\text{ and }b\text{ are Orthogonal}$